Mathematics 244, section 1 – Linear Algebra Sample Exam 2 – March 16, 2007

I. All parts of this question refer to the matrix

A = ( \begin{matrix}1 & 0 & 2 & 3 & −1 \\ 2 & 1 & −1 & 1 & s \\ −1 & −1 & 3 & 2 & −1 \end{matrix} )

A) For which value of the scalar s does A satisfy rank(A) = 2? In the next two parts use

this value of s. B) Give a basis of Col(A). C) Find the dimension of Nul(A) and give a basis of Nul(A).

II. A) Use Cramer’s Rule to solve

\begin{aligned}5x_{1} + 3x_{2} − x_{3} & = 1 \\ 3x_{1} + x_{2} + x_{3} & = 2 \\ x_{1} + x_{2} − x_{3} & = 0 \end{aligned}

B) Prove Cramer’s Rule: If A is an invertible n × n matrix, then the solution of Ax = b

is the vector x with entries

x_{i} = \frac{det(A_{i}(b))}{det(A)}

for i = 1, …, n.

III. Let V = P_{3}(R), the vector space of all polynomials of degree ≤ 3 with real coefficients. A) Is T : V → V defined by T (p) = p′ − 2p a linear transformation (p′ = derivative of

p)? Why or why not? B) Is

W = {p ∈ V : p′(x) − 2p(x) = 0 for all x}

a vector subspace of V? Why or why not?

IV. (True-False) For each true statement, give a short proof. For each false statement, give a counterexample. (Note: As always, “true” means “true in every case.”) A) If T : R^{5} → R^{3} is linear, A is the standard matrix of T, and S = {v_{1}, v_{2}, v_{3}} ⊂ Nul(A),

then S is linearly dependent. B) Let A, B be n × n matrices. If det(A) = −3 and B^{2} = I, then B^{t}AB is an invertible

matrix. C) If A is an m × n matrix, then rank(A) + dim Nul(A) = n.